Optimizing (1-x) log(1+(g(x)/h(x)))

My immediately preceding post addressed the -t*log2(...) term., which answered a post you subsequently deleted.

The T*log2(...) term is also concave, and i think can be handled by adapting the solution at Can CVX solve this kind of function {x-log(1+0.01*x/(x+1))} . I leave the details of the adapting to you.